The evolution of the optical field for an optical klystron(OK) free electron laser is described analytically for both low gain case and high gain case. The harmonic optical klystron (HOK) in which the second undulator is resonant on the higher harmonic of the first undulator is analysed as a harmonic amplifier. The optical field evolution equation of the HOK is derived analytically for both CHG mode (Coherent Harmonic Generation, the quadratic gain regime) and HGHG mode (High Gain Harmonic Generation, the exponential gain regime), the effects of energy spread, energy modulation and dispersion in the whole process are considered. For the quadratic gain regime, the analytical formula of CHG mode is given and compared with that of OK case. For the exponential gain regime, the linear theory of HGHG mode is given and discussed.

The high-gain harmonic generation (HGHG) free-electron laser (FEL) is an important candidate for a fourth-generation light source. Lots of theoretical work has been performed. Recently a further 1D theory about HGHG FEL has been developed. It considers the effects of different parameters for the whole process. An initial program based on this theory has been made. In this paper, a brief comparison of the results from this 1D program and from TDA (3D code) is discussed. It also analyses the parameters for Shanghai deep ultra violate free-electron laser source (SDUV-FEL), including electron beam energy spread, seed laser power, strength of dispersion section etc.

The design study of the undu1ator for Shanghai deep ultra violate free electron laser source (SDUV-FEL) is presented. The optimum undulator parameters for the FEL performance have been studied. The scheme of focusing and segmentation is discussed. The requirements of undulator magnet field and main technical demand are given.